When trying to solve
arctan(sqrt 567.2 / -459.6) my answer was -2.96. My teacher said that i handled the radical wrong. She said I have to square both numbers, then add them, and then find the square root. I am confused because you cannot take the square root of -459.6. Please explain.

most calculators have been programmed that for any arc(trig function) they will give you the smallest value of rotation from 0,
so -2.97 would be in the 4th quadrant, coterminal with 360-2.97 or 357.03°

the tangent is also negative in the 2nd quadrant, so
the smallest positive answer is 180-2.97 = 177.03°